Kirkwood-Buff integrals: From fluctuations in finite volumes to the thermodynamic limit

The Kirkwood-Buff theory is a cornerstone of the statistical mechanics of liquids and solutions. It relates volume integrals over the radial distribution function, so-called Kirkwood-Buff integrals (KBIs), to particle number fluctuations and thereby to various macroscopic thermodynamic quantities such as the isothermal compressibility and partial molar volumes. Recently, the field has seen a strong revival with breakthroughs in the numerical computation of KBIs and applications to complex systems such as bio-molecules. One of the main emergent results is the possibility to use the finite volume KBIs as a tool to access finite volume thermodynamic quantities. The purpose of this Perspective is to shed new light on the latest developments and discuss future avenues.

A procedure to find thermodynamic equilibrium constants for CO2 and CH4 adsorption on activated carbon

Thermodynamic equilibrium for adsorption means that the chemical potential of gas and adsorbed phase are equal. A precise knowledge of the chemical potential is, however, often lacking, because the activity coefficient of the adsorbate is not known. Adsorption isotherms are therefore commonly fitted to ideal models such as the Langmuir, Sips or Henry models. We propose here a new procedure to find the activity coefficient and the equilibrium constant for adsorption which uses the thermodynamic factor. Instead of fitting the data to a model, we calculate the thermodynamic factor and use this to find first the activity coefficient. We show, using published molecular simulation data, how this procedure gives the thermodynamic equilibrium constant and enthalpies of adsorption for CO2(g) on graphite. We also use published experimental data to find similar thermodynamic properties of CO2(g) and of CH4(g) adsorbed on activated carbon. The procedure gives a higher accuracy in the determination of enthalpies of adsorption than ideal models do.

Calculation of the chemical potential and the activity coefficient of two layers of CO2 adsorbed on a graphite surface

Thermodynamics of two layers of CO₂ on a graphite surface obtained directly from the simulations and the Small System Method.

Reciprocal relations in electroacoustics

In a colloidal suspension, one can generate sound waves by the application of an alternating electric field (Electrokinetic Sonic Amplitude, i.e., ESA). Another phenomenon is electrophoresis (Electrophoretic Mobility, i.e., EM) where a colloidal particle moves relative to the solvent in an electric field. Vice versa one can generate electric fields or electric currents by sound waves (Colloid Vibration Potential/Current, i.e., CVP/CVI). In 1988 and 1990, O'Brien [J. Fluid Mech. 190, 71-86 (1988) and O'Brien, J. Fluid Mech. 212, 81-93 (1990)] derived a reciprocal relation between the proportionality coefficients of the EM and CVI phenomena. In this paper, we will generalize his proof by constructing the relevant entropy production from which the linear force-flux relations follow. General relations are derived for electrolyte solutions, of which colloidal suspensions are a particular case. The relations between CVI, CVP, EM, and ESA are discussed. O'Brien's reciprocal relation then follows as an Onsager relation. The relation is valid for any applied electric field frequency, particle surface charge and particle concentration (even in the presence of particle-particle interactions) provided the system is isotropic.

Mesoscopic non-equilibrium thermodynamic analysis of molecular motors

We show that the kinetics of a molecular motor fueled by ATP and operating between a deactivated and an activated state can be derived from the principles of non-equilibrium thermodynamics applied to the mesoscopic domain. The activation by ATP, the possible slip of the motor, as well as the forward stepping carrying a load are viewed as slow diffusion along a reaction coordinate. Local equilibrium is assumed in the reaction coordinate spaces, making it possible to derive the non-equilibrium thermodynamic description. Using this scheme, we find expressions for the velocity of the motor, in terms of the driving force along the spacial coordinate, and for the chemical reaction that brings about activation, in terms of the chemical potentials of the reactants and products which maintain the cycle. The second law efficiency is defined, and the velocity corresponding to maximum power is obtained for myosin movement on actin. Experimental results fitting with the description are reviewed, giving a maximum efficiency of 0.45 at a myosin headgroup velocity of 5 × 10(-7) m s(-1). The formalism allows the introduction and test of meso-level models, which may be needed to explain experiments.

Curvature dependence of the interfacial heat and mass transfer coefficients

Nucleation is often accompanied by heat transfer between the surroundings and a nucleus of a new phase. The interface between two phases gives an additional resistance to this transfer. For small nuclei the interfacial curvature is high, which affects not only equilibrium quantities such as surface tension, but also the transport properties. In particular, high curvature affects the interfacial resistance to heat and mass transfer. We develop a framework for determining the curvature dependence of the interfacial heat and mass transfer resistances. We determine the interfacial resistances as a function of a curvature. The analysis is performed for a bubble of a one-component fluid and may be extended to various nuclei of multicomponent systems. The curvature dependence of the interfacial resistances is important in modeling transport processes in multiphase systems.

Effect of compressibility in bubble formation in closed systems

We analyze the stability of small bubbles in a closed system with fixed volume, temperature, and number of molecules. We show that there exists a minimum stable size of a bubble. Thus there exists a range of densities where no stable bubbles are allowed and the system has a homogeneous density which is lower than the coexistence density of the liquid. This becomes possible due to the finite liquid compressibility. Capillary analysis within the developed "modified bubble" model illustrates that the existence of the minimum bubble size is associated to the compressibility and it is not possible when the liquid is strictly incompressible. This finding is expected to have very important implications in cavitation and boiling.

The role of temperature in nucleation processes

Heat and mass transfers are coupled processes, also in nucleation. In principle, a nucleating cluster would have a different temperature compared to the surrounding supersaturated old phase because of the heat release involved with attaching molecules to the cluster. In turn a difference in temperature across the cluster surface is a driving force for the mass transfer to and from the cluster. This coupling of forces in nonisothermal nucleation is described using mesoscopic nonequilibrium thermodynamics, emphasizing measurable heat effects. An expression was obtained for the nonisothermal nucleation rate in a one-component system, in the case where a temperature difference exists between a cluster distribution and the condensed phase. The temperature is chosen as a function of the cluster size only, while the temperature of the condensed phase is held constant by a bath. The generally accepted expression for isothermal stationary nucleation is contained as a limiting case of the nonisothermal stationary nucleation rate. The equations for heat and mass transport were solved for stationary nucleation with the given cluster distribution and with the temperature controlled at the boundaries. A factor was defined for these conditions, determined by the heat conductivity of the surrounding phase and the phase transition enthalpy, to predict the deviation between isothermal and nonisothermal nucleation. For the stationary state described, the factor appears to give small deviations, even for primary nucleation of droplets in vapor, making the nonisothermal rate smaller than the isothermal one. The set of equations may lead to larger and different thermal effects under different boundary conditions, however.

Numerical solution of the nonequilibrium square-gradient model and verification of local equilibrium for the Gibbs surface in a two-phase binary mixture

In this paper we apply the general analysis described in our first paper to a binary mixture of cyclohexane and n -hexane. We use the square gradient model for the continuous description of a nonequilibrium surface and obtain numerical profiles of various thermodynamic quantities in various stationary state conditions. Details of the numerical procedure are given and discussed. In the second part of this paper we focus on the verification of local equilibrium of the surface as described with excess densities. We give a definition of the temperature and chemical potentials of the surface and verify that these quantities are independent of the choice of the dividing surface. We verify numerically that the surface in a stationary state of the mixture can be described in terms of Gibbs excess densities, which are found to be in good approximation equal to their equilibrium values at the stationary state temperature and chemical potentials of the surface.

Nonequilibrium properties of a two-dimensionally isotropic interface in a two-phase mixture as described by the square gradient model

In earlier work a systematic extension of the van der Waals square gradient model to nonequilibrium one-component systems was given. In this work the focus was on heat and mass transfer through the liquid-vapor interface as caused by a temperature difference or an over- or underpressure. We will give an extension of this approach to multicomponent nonequilibrium systems in the systematic context of nonequilibrium thermodynamics. An explicit expression for the pressure tensor is derived valid also for curved surfaces. It is shown how the Gibbs relation should be modified in the interfacial region, in both equilibrium and nonequilibrium. The two-dimensional isotropy of a curved interface is discussed. Furthermore, we give numerically obtained profiles of the concentration, the mole fraction, and the temperature, which illustrate the solution for some special cases.

Thermal effects during adsorption of n-butane on a silicalite-1 membrane: A non-equilibrium molecular dynamics study

Non-equilibrium molecular dynamic (NEMD) simulations have been used to study the kinetics of adsorption of n-butane molecules in a silicalite membrane. We have chosen this simple well-known process to demonstrate that the process is characterized by two stages, both non-isothermal. In the first stage the large chemical driving force leads to a rapid uptake of n-butane in all the membrane and a simultaneous increase in the membrane temperature, explained by the large enthalpy of adsorption, DeltaH=-61.6kJ/mol butane. A diffusion coefficient for transport across the external surface layer is calculated from the relaxation time; a value of 3.4x10(-9)m(2)/s is found. During the adsorption, a significant thermal driving force develops across the external surface of the membrane, which leads to an energy flux out of the membrane during the second stage. In this stage a thermal conductivity of 3.4x10(-4)W/Km is calculated from the corresponding relaxation time for the surface, confirming that the thermal conduction is the rate-limiting step. The aim of this paper is to demonstrate that a thermal driving force must be taken into account in addition to a chemical driving force in the description of transport in nano-porous materials.

Thermodynamics for single-molecule stretching experiments

We show how to construct nonequilibrium thermodynamics for systems too small to be considered thermodynamically in a traditional sense. Through the use of a nonequilibrium ensemble of many replicas of the system which can be viewed as a large thermodynamic system, we discuss the validity of nonequilibrium thermodynamics relations and analyze the nature of dissipation in small systems through the entropy production rate. We show in particular that the Gibbs equation, when formulated in terms of average values of the extensive quantities, is still valid, whereas the Gibbs-Duhem equation differs from the equation obtained for large systems due to the lack of the thermodynamic limit. Single-molecule stretching experiments are interpreted under the prism of this theory. The potentials of mean force and mean position, now introduced in these experiments in substitution of the thermodynamic potentials, correspond respectively to our Helmholtz and Gibbs energies. These results show that a thermodynamic formalism can indeed be applied at the single-molecule level.

Unifying Thermodynamic and Kinetic Descriptions of Single-Molecule Processes: RNA Unfolding under Tension

We use mesoscopic nonequilibrium thermodynamics theory to describe RNA unfolding under tension. The theory introduces reaction coordinates, characterizing a continuum of states for each bond in the molecule. The unfolding considered is so slow that one can assume local equilibrium in the space of the reaction coordinates. In the quasi-stationary limit of high sequential barriers, our theory yields the master equation of a recently proposed sequential-step model. Nonlinear switching kinetics is found between open and closed states. Our theory unifies the thermodynamic and kinetic descriptions and offers a systematic procedure to characterize the dynamics of the unfolding process.

Transfer coefficients for evaporation of a system with a Lennard-Jones long-range spline potential

Surface transfer coefficients are determined by nonequilibrium molecular dynamics simulations for a Lennard-Jones fluid with a long-range spline potential. In earlier work [A. Røsjorde, J. Colloid Interface Sci. 240, 355 (2001); J. Xu, ibid. 299, 452 (2006)], using a short-range Lennard-Jones spline potential, it was found that the resistivity coefficients to heat and mass transfer agreed rather well with the values predicted by kinetic theory. For the long-range Lennard-Jones spline potential considered in this paper we find significant discrepancies from the values predicted by kinetic theory. In particular the coupling coefficient, and as a consequence the heat of transfer on the vapor side of the surface are much larger. Thermodynamic data for the liquid-vapor equilibrium confirmed the law of corresponding states for the surface, when it is described as an autonomous system. The importance of these findings for modelling phase transitions is discussed.

Interface Film Resistivities for Heat and Mass TransfersIntegral Relations Verified by Non-equilibrium Molecular Dynamics

Integral relations that predict interface film transfer coefficients for evaporation and condensation have recently been derived. According to these relations, all coefficients can be calculated for one-component systems, using the thermal resistivity and the enthalpy profile through the interface. The integral relations were tested in this work using nonequilibrium molecular dynamics simulations for argon-like particles and n-octane molecules. The simulations confirm the integral relations within the accuracy of the calculation for both systems. Evidence is presented for the existence of an excess thermal resistivity on the gas side of the surface, and the fact that this property is decisive for interface heat and mass transfer coefficients. The integral relations were used to predict the mass transfer coefficient for n- octane as a function of surface tension. The findings are important for modeling of one-component phase transitions.

Verification of Onsager's reciprocal relations for evaporation and condensation using non-equilibrium molecular dynamics

Non-equilibrium molecular dynamic (NEMD) simulations have been used to study heat and mass transfer across a vapor-liquid interface for a one-component system using a Lennard-Jones spline potential. It was confirmed that the relation between the surface tension and the surface temperature in the non-equilibrium system was the same as in equilibrium (local equilibrium). Interfacial transfer coefficients were evaluated for the surface, which expressed the heat and mass fluxes in temperature and chemical potential differences across the interfacial region (film). In this analysis it was assumed that the Onsager reciprocal relations were valid. In this paper we extend the number of simulations such that we can calculate all four interface film transfer coefficients along the whole liquid-vapor coexistence curve. We do this analysis both for the case where we use the measurable heat flux on the vapor side and for the case where we use the measurable heat flux on the liquid side. The most important result we found is that the coupling coefficients within the accuracy of the calculation are equal. This is the first verification of the validity of the Onsager relations for transport through a surface using molecular dynamics. The interfacial film transfer coefficients are found to be a function of the surface temperature alone. New expressions are given for the kinetic theory values of these coefficients which only depend on the surface temperature. The NEMD values were found to be in good agreement with these expressions.

Electrically induced anisotropy in a colloidal dispersion of nanospheres as measured by electric birefringence

Electrically induced birefringence experiments were performed on dispersions consisting of sulfate latex nanospheres of two different sizes and charges dispersed in an electrolyte solution, at various ionic strengths. The induced birefringence was found to have an important contribution increasing as a quadratic power law of the volume fraction of the spheres. This shows that interparticle interactions play a role in the observed birefringence. The data were analyzed, using a theory from Hafkenscheid and Vlieger [Physica 75 (1974) 57], in terms of the changes of the interparticle separations in the directions parallel and perpendicular to the applied electric field.

Active transport: a kinetic description based on thermodynamic grounds

We show that active transport processes in biological systems can be understood through a local equilibrium description formulated at the mesoscale, the scale to describe stochastic process. This new approach uses the method established by nonequilibrium thermodynamics to account for the irreversible processes occurring at this scale and provides nonlinear kinetic equations for the rates in terms of the driving forces. The results show that the application domain of nonequilibrium thermodynamics method to biological systems goes beyond the linear domain. A model for transport of Ca2+ by the Ca2+-ATPases, nonlinear way. Our results unify thermodynamic and kinetic descriptions, thereby opening new perspectives in the study of different transport phenomena in biological systems.

Nonequilibrium thermodynamic description of the three-phase contact line

In order to give a solid foundation of the description of the motion of the three-phase contact line, we develop the nonequilibrium thermodynamic description of the contact line. It is postulated that during its motion the contact line is featured as a separate thermodynamic system. Conservation laws are given for the excess densities of the components, the momentum and the energy along the line. The Gibbs law is formulated for the contact line and using this law the excess entropy production density along the line is constructed. This identifies the conjugate thermodynamic forces and fluxes for the contact line. Linear laws relating these quantities can then be given. The special case considered by Shikmurzaev, who gave the first satisfactory description of the motion of the contact line, is considered in more detail. A new boundary condition is found, which was not used by Shikhmurzaev. The need for such an additional boundary condition for this case was discussed in recent work by Billingham.

The interpretation of dielectric spectroscopy measurements on silica and hematite sols

Experimental data on the dielectric response of silica and hematite sols from the literature are interpreted using an analytical theory developed previously (Chassagne, C., Bedeaux, D., and Koper, G. J. M., J. Phys. Chem B105, 11,743 (2001), and Physica A, to be published). The theory is found to correctly predict both the relaxation frequency and the dielectric permittivity enhancement at low frequencies with only one free parameter. This parameter can be the zeta potential or the Stern layer conductance, in the case that the zeta potential is fixed to agree with the electrophoretic mobility measurements.

Nonequilibrium Molecular Dynamics Simulations of Steady-State Heat and Mass Transport in Condensation. II. Transfer Coefficients

We present coefficients for transfer of heat and mass across the liquid-vapor interface of a one-component fluid. The coefficients are defined for the Gibbs surface from nonequilibrium thermodynamics and determined by nonequilibrium molecular dynamics simulations. The main conductivity coefficients are found to become large near the critical point, consistent with the disappearance of the surface in this limit. The resistivities of transfer found by molecular dynamics simulations are compared to the values predicted by kinetic theory. The main resistivity to heat transfer is found to agree from the triple point to about halfway to the critical point. The resistivity to mass transfer was used to determine the condensation coefficient, which was found to be practically constant with a value of about 0.82. The resistivity coupling coefficient predicted by simulations also agrees with values predicted by kinetic theory from the triple point until about halfway to the critical point. Copyright 2001 Academic Press.

Osmotic Compressibility of Poly(propylene imine) Dendrimers in Deuterated Methanol

The inverse osmotic compressibility of the poly(propylene imine) dendrimers in deuterated methanol has been measured as a function of concentration with small-angle neutron scattering. The experimental results reveal three different regimes. First, there is a dilute regime going up to a maximum in the inverse osmotic compressibility. This region can be subdivided into a very dilute region, where the behavior is hard-sphere-like, and a denser region, where the solvation layers overlap. The maximum, occurring around volume fraction 0.30 for each generation, is found to be the concentration where the distance between the centers of two dendrimers is twice their radius of gyration. It designates the crossover to the second regime of a semidilute phase with shrinking dendrimers. Interpenetration of the dendrimers does not seem to take place. Finally, for the higher generations, at high concentrations, the dendrimers are collapsed and the inverse osmotic compressibility starts to increase again. As dendrimers are regularly and very highly branched molecules, they can be considered as ultimately hyperbranched polymers. For both types, the experimental inverse osmotic compressibility shows similar features. The dendrimers seem to be more compact, however. Copyright 2000 Academic Press.

Nonequilibrium Molecular Dynamics Simulations of Steady-State Heat and Mass Transport in Condensation

We present evidence for the hypothesis of local equilibrium for a liquid-vapor interface in a one-component fluid, using molecular dynamics simulations. Lennard-Jones/spline particles are studied in a two-phase system that is out of global equilibrium. Equilibrium simulations are first used to establish the equation of state for the vapor and interface. A procedure is developed to define the boundaries of the interface. Finally it is shown that the equation of state for the interface applies also when there is heat and mass transport through the interface. The temperature gradient in the vapor was 10(8) K/m in these studies. Copyright 2000 Academic Press.

Impedance of the Hydrogen Polymer Fuel Cell Electrode. Theory and Experiments

We derive the impedance for the hydrogen electrode in the polymer membrane fuel cell from irreversible thermodynamics. The results predict a surface contribution to the cell impedance that can give two semi-circles in the Nyquist diagram. The equivalent circuit of the impedance is shown. The high-frequency contribution is connected to the oscillation of dipoles consisting of free charges in the surface, while the low-frequency contribution is connected to the electrochemical reaction. This can be explained by a slowly relaxing proton conducting polymer network at the reaction site.

A molecular theory for nonohmicity of the ion leak across the lipid-bilayer membrane

The current-voltage relationship of ion leak (i.e., ion transport involving neither special channels nor carriers) across the lipid-bilayer membrane has been observed to be log-linear above the ohmic regime. The coefficient of the linear term has been found to be universal for membranes and penetrants examined. This universality has been explained in terms of diffusion in an external field, where the ion position is described as a Markovian process. Such a diffusion picture can be questioned, however. It is also probable that a leaking ion gets over the potential barrier before experiencing sufficient random collision in the membrane, considering that each ion is surrounded with long lipid molecules aligned almost unidirectionally. As an alternative, we discuss this ion leak in terms of velocity distribution of the ions entering the membrane and density fluctuation of the lipids. We conclude that we can explain the universality without resorting to the diffusion picture.